An introduction to the gauge integral (also known as the Henstock, Kurzweil, or
generalized Riemann integral), comparing it with the Riemann and Lebesgue integrals. Historical and Bibliographical Overview; The Definition; Simplicity and Concreteness of the Definitions; A Simple Example; Generality: Existence of Integrals; Fundamental Theorems; Improper Integrals; Convergence Theorems; The role of the gauge integral in teaching analysis; Some remarks on the lack of pathological examples; Some Open Research Problems.